Electrical Mobility as an Indicator for Flexibly Deducing the Kinetics of Nanoparticle Evaporation

Condensation and evaporation of vapor species on nanoparticle surfaces drive the aerosol evolution in various industrial/atmospheric systems, but probing these transient processes is challenging due to related time and length scales. Herein, we present a novel methodology for deducing nanoparticle evaporation kinetics using electrical mobility as a natural size indicator. Monodispersed nanoparticles are fed to a differential mobility analyzer which serves simultaneously as an evaporation flowtube and an instrument for measuring the electrical mobility, realizing measurements of evaporation processes with time scales comparable to the instrument response time. A theoretical framework is derived for deducing the evaporation kinetics from instrument responses through analyzing the nanoparticle trajectory and size–mobility relationship, which considers the coupled mass and heat transfer effect and is applicable to the whole Knudsen number range. The methodology is demonstrated against evaporation but can potentially be extended to condensation and other industrial/atmospheric processes involving rapid size change of nanoparticles.


Theoretical derivations for operations in the free molecular and continuum regime
The derivation of the glycerol nanoparticle sizes at the DMA outlet based on measured nominal sizes for evaporation in the free molecular and continuum regime follows a similar procedure as that of the transition regime. Here, we show the theoretical methods for the free molecular and continuum regimes separately. The same set of notations are used as that of the main text.
Continuum regime. The flux in the continuum regime is: with, ). (S1b) The relationship between the radius and mobility is: Substituting Eqs. S1a -S2 into Eq. 3 in the main text, one obtains: with, the solution of which is: Substitution of Eq. S4 into Eq. 2 in the main text yields: Note that this result is the same as the result in the transition regime because of the form for the expression of corresponding fluxes (Eq. S1a and Eq. 4a). But Eqs. S5a and S5b should have higher precisions than Eqs. 8a and 8b when invoked in the continuum regime because one does not resort to the assumption that ( v )/ 2 ( g ) ≈ ( v ̅̅̅̅̅ )/ 2 ( g ̅̅̅̅̅ ) in reaching Eqs. S5a and S5b.
Free molecular regime. The flux in the free molecular regime is: and being the mean thermal speed of vapor molecules, where v is the vapor molecular mass. The relationship between the radius and mobility is: where, = 1.36 is the momentum transfer accommodation coefficient 1 , g is the background gas molecular weight, g is the background gas concentration, and g and is the background gas mean thermal speed. Substituting Eqs. S6a -S7 into Eq. 3 in the main text, one obtains: with the solution of which is: Substitution of Eq. S9 into Eq. 2 in the main text yields: is the glycerol density, the mass accommodation coefficient was assumed to be unit = 1 for the solid line and 0.6 < < 1 for the shadowed region in Fig. 2, and the sheath flow was assumed to be dry air (in consistent with the experimental conditions) such that ∞ = 0.

Simulation of the evaporation of a free glycerol nanoparticle and related material properties
For water: properties are obtained online at https://www.engineeringtoolbox.com/. Figure S1. Size evolution of free glycerol and water particles at selected temperatures with or without heat transfer.

Effects of the approximations made in Eq. 6b of the main text
The simulation procedure for obtaining nanoparticle mobilities at the DMA outlet (solid line) and nominal mobilities (dotted line) in the figure has been described in the previous section.

Influence of the diffusion coefficient
The diffusion coefficient essentially determines the flux of evaporation, thus it is helpful to check its influence on the calculations. The value of the diffusion coefficient influences the numerical simulated results of the glycerol nanoparticle radius at the DMA outlet but will not influence the theoretically deduced values, as the latter is lumped in the constant which is derived from the operation conditions of the device and the experimentally measured nominal radius. We increased and decreased the value of the diffusion coefficient used in the main text by 50% respectively, to mimic possible errors in the estimation. The results for the simulated glycerol nanoparticle radius at the DMA outlet are shown in Fig. S5 for the case of sh = 26.48 L min -1

S9
and pi = 151 nm. Overall, this range leads to results that overlap well with the deduced results, indicating that the diffusion coefficient value adopted is reasonable. Figure S5. Influence of the diffusion coefficient on the simulated glycerol nanoparticle radius at the DMA outlet for the case of sh = 26.48 L min -1 and pi = 151 nm.